In normal distribution, the probability density function of random variable X is a member of family of function

\(f(x)=\dfrac {1}{\sigma\sqrt {2\pi}}e\;^{-(x-\mu)^2/2\sigma^2}\)

where,

\(\mu\)= mean of distribution

\(\sigma\) = standard deviation of distribution

In this case,

\(f(x)=\dfrac {1}{20\sqrt {2\pi}}e\;^{-(x-100)^2/2×20^2}\)

Comparing with standard function we get

\(\mu=\) 100, \(\sigma=\)20

The Normal distribution, density function of a random variable X is of the function
\(f(x)=\dfrac {1}{20\sqrt {2\pi}}\,e\,^{-(x-100)^2/2×20^2}\)
Find the value of \(\mu\) and \(\sigma\) for this distribution.

It is found that height of adult males in united states are normally distributed with means 69 inches and standard deviation 2.8 inches what is the probability that an adult male chosen at random is between 70 and 75 inches tall. (Just state the interval).

A

\(\int\limits_{70}^{75}\dfrac {1}{10}e\;^{-(x-100)^2/2}\; dx\)

.

B

\(\int\limits_{5}^{10} e\;^{-x^2}\; dx\)

C

\(\int\limits_{70}^{75}\dfrac {1}{7.02}e\;^{-(x-63)^2/15.68}\; dx\)

D

\(\int\limits_{1}^{2}e\;^-x^2\; dx\)

Option C is Correct

Cumulative Distribution Function (CDF)

Let \(X\) define a continuous random variable then we define cumulative distribution function of this random variable as

\(P(X\leq x)=\displaystyle\int \limits^x_{-\infty}f(x)\,dx=F(x)\)

It is basically the probability that the random variable \(X\) will take value less than \(x\).

A random variable \(X\) has the probability density function
\(f(x)=\dfrac{1}{\pi(x^2+1)}\) for all \(x\,\varepsilon R\).
Find the value of \(F(1)\) where \(F(x)\) is the cumulative distribution function.

A

\(\dfrac{5}{6}\)

.

B

\(\dfrac{3}{4}\)

C

\(\dfrac{1}{2}\)

D

\(\dfrac{5}{7}\)

Option B is Correct

Speed of Vehicles on the highway (Normal distribution)

It is observed that speed of vehicles which run on a highway are normally distributed and if the maximum speed limits is known, the probability that a randomly chosen vehicle is traveling at a legal speed can be found.

The speed of vehicles on a highway with speed limit 100 km/hr are normally distributed with mean = \(\mu\)= 110 km/hr and standard deviation \(\sigma=\)10 km/hr. What is the probability that a randomly chosen vehicle is traveling at a legal speed.

A

.7963

.

B

.1584

C

1.0832

D

.0014

Option B is Correct

Probability of the Random Variable lies within n Standard Deviation About the Mean in Normal Distribution Function

Consider the probability density function for random variable

\(f(x)=\dfrac {1}{\sigma \sqrt{2\pi}}\,e^{-(x-\mu)^2/2\sigma^2}\)

Suppose we want to know what is the probability that random variable X lie within. one standard deviation of the mean i.e.